Merlin-Arthur with efficient quantum Merlin and quantum supremacy for the second level of the Fourier hierarchy

Tomoyuki Morimae1,2,3, Yuki Takeuchi4,5, and Harumichi Nishimura6

1Yukawa Institute for Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan
2Department of Computer Science, Gunma University, 1-5-1 Tenjincho, Kiryu, Gunma, 376-0052, Japan
3JST, PRESTO, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012, Japan
4Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
5NTT Communication Science Laboratories, NTT Corporation, 3-1 Morinosato Wakamiya, Atsugi, Kanagawa 243-0198, Japan
6Graduate School of Informatics, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi, 464-8601, Japan

We introduce a simple sub-universal quantum computing model, which we call the Hadamard-classical circuit with one-qubit (HC1Q) model. It consists of a classical reversible circuit sandwiched by two layers of Hadamard gates, and therefore it is in the second level of the Fourier hierarchy. We show that output probability distributions of the HC1Q model cannot be classically efficiently sampled within a multiplicative error unless the polynomial-time hierarchy collapses to the second level. The proof technique is different from those used for previous sub-universal models, such as IQP, Boson Sampling, and DQC1, and therefore the technique itself might be useful for finding other sub-universal models that are hard to classically simulate. We also study the classical verification of quantum computing in the second level of the Fourier hierarchy. To this end, we define a promise problem, which we call the probability distribution distinguishability with maximum norm (PDD-Max). It is a promise problem to decide whether output probability distributions of two quantum circuits are far apart or close. We show that PDD-Max is BQP-complete, but if the two circuits are restricted to some types in the second level of the Fourier hierarchy, such as the HC1Q model or the IQP model, PDD-Max has a Merlin-Arthur system with quantum polynomial-time Merlin and classical probabilistic polynomial-time Arthur.

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Cited by

[1] François Le Gall, Tomoyuki Morimae, Harumichi Nishimura, and Yuki Takeuchi, "Interactive Proofs with Polynomial-Time Quantum Prover for Computing the Order of Solvable Groups", arXiv:1805.03385 (2018).

[2] Mariami Gachechiladze, Otfried Gühne, and Akimasa Miyake, "Changing the circuit-depth complexity of measurement-based quantum computation with hypergraph states", arXiv:1805.12093 (2018).

[3] Keisuke Fujii, "Quantum speedup in stoquastic adiabatic quantum computation", arXiv:1803.09954 (2018).

[4] Tomoyuki Morimae and Harumichi Nishimura, "Rational proofs for quantum computing", arXiv:1804.08868 (2018).

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